Matchings of quadratic size extend to long cycles in hypercubes

TL;DR

This paper proves that in an n-dimensional hypercube, any matching of quadratic size can be extended to a cycle covering at least three-quarters of the vertices, advancing understanding of cycle extensions in hypercubes.

Contribution

It introduces a quadratic bound for matchings in hypercubes that guarantees extension to large cycles covering at least 75% of vertices.

Findings

Matchings of size up to q(n) extend to large cycles

Cycle coverage is at least 75% of vertices

Advances the extension problem in hypercubes

Abstract

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.